scholarly journals Mixed methods for degenerate elliptic problems and application to fractional Laplacian

2020 ◽  
Author(s):  
María Eugenia Cejas ◽  
Ricardo Durán ◽  
Mariana Prieto
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Mixed Finite Element ◽  
Optimal Order ◽  
Order Error ◽  
Quadrilateral Elements ◽  
Muckenhoupt Class ◽  
Degenerate Elliptic ◽  
Weighted Norms ◽  
Solutions Of Equations

  We analyze the approximation by mixed finite element methods of solutions of     equations of the form div  [[EQUATION]]  , where the coefficient a=a(x) can     degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the     coefficient $a$ belongs to the Muckenhoupt class  [[EQUATION]] .     The analysis developed applies to general mixed finite element spaces satisfying the     standard commutative diagram property, whenever some stability and interpolation     error estimates are valid in weighted norms. Next, we consider in detail the case     of Raviart-Thomas spaces of arbitrary order, obtaining optimal order error estimates for simplicial elements in any dimension and for convex quadrilateral elements in the two dimensional case, in both cases under a regularity assumption on the family of meshes.          For the lowest order case we show that the regularity assumprtion can be removed and prove  anisotropic error estimates which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.

scholarly journals Analysis of finite element methods for vector Laplacians on surfaces

2019 ◽  
Vol 40 (3) ◽  
pp. 1652-1701 ◽  
Author(s):  
Peter Hansbo ◽  
Mats G Larson ◽  
Karl Larsson
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Vector Fields ◽  
Optimal Order ◽  
Order Error ◽  
Optimal Order Error Estimates ◽  
Elastic Membranes ◽  
Tangential Vector ◽  
Higher Order Finite Elements ◽  
Lagrange Elements

Abstract We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in ${\mathbb{R}}^3$. Closely related operators arise in models of flow on surfaces as well as elastic membranes and shells. The method is based on standard continuous parametric Lagrange elements that describe a ${\mathbb{R}}^3$ vector field on the surface, and the tangent condition is weakly enforced using a penalization term. We derive error estimates that take into account the approximation of both the geometry of the surface and the solution to the partial differential equation. In particular, we note that to achieve optimal order error estimates, in both energy and $L^2$ norms, the normal approximation used in the penalization term must be of the same order as the approximation of the solution. This can be fulfilled either by using an improved normal in the penalization term, or by increasing the order of the geometry approximation. We also present numerical results using higher-order finite elements that verify our theoretical findings.

scholarly journals A New Characteristic Nonconforming Mixed Finite Element Scheme for Convection-Dominated Diffusion Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Dongyang Shi ◽  
Qili Tang ◽  
Yadong Zhang
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Auxiliary Variable ◽  
Diffusion Problem ◽  
Optimal Order ◽  
Order Error ◽  
Elliptic Projection ◽  
Indispensable Tool ◽  
Convection Dominated

A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variableuand the auxiliary variableσwith respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results to confirm the theoretical analysis.

scholarly journals Expanded Mixed Finite Element Method for the Two-Dimensional Sobolev Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Qing-li Zhao ◽  
Zong-cheng Li ◽  
You-zheng Ding
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Two Dimensional ◽  
Sobolev Equation ◽  
Order Error ◽  
Expanded Mixed Finite Element ◽  
Element Method

Expanded mixed finite element method is introduced to approximate the two-dimensional Sobolev equation. This formulation expands the standard mixed formulation in the sense that three unknown variables are explicitly treated. Existence and uniqueness of the numerical solution are demonstrated. Optimal order error estimates for both the scalar and two vector functions are established.

Numerical Analysis for Two-phase Flow in Porous Media

2003 ◽  
Vol 3 (1) ◽  
pp. 59-75
Author(s):  
Zhangxin Chen
Keyword(s):  
Finite Element ◽  
Porous Media ◽  
Error Estimates ◽  
Finite Element Methods ◽  
Compressible Flows ◽  
Mixed Finite Element ◽  
Flow In Porous Media ◽  
Optimal Order ◽  
Two Phase ◽  
Characteristic Finite Element

Abstract In this paper we derive error estimates for finite element approximations for partial differential systems which describe two-phase immiscible flows in porous media. These approximations are based on mixed finite element methods for pressure and velocity and characteristic finite element methods for saturation. Both incompressible and compressible flows are considered. Error estimates of optimal order are obtained.

Error Estimates of H1-Galerkin Expanded Mixed Finite Element Methods for Heat Problems

2011 ◽  
Vol 267 ◽  
pp. 493-498
Author(s):  
Hai Tao Che ◽  
Mei Xia Li ◽  
Li Juan Liu
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Low Permeability ◽  
Heat Equations ◽  
Order Error ◽  
Mixed Element ◽  
Expanded Mixed Finite Element ◽  
Element Method

H1-Galerkin expanded mixed element method are discussed for a class of second-order heat equations. The methods possesses the advantage of mixed finite element while avoiding directly inverting the permeability tensor, which is important especially in a low permeability zone. H1-Galerkin expanded mixed finite element method for heat equations are described, an optimal order error estimate for the methods is obtained.

H1 -Galerkin Expanded Mixed Finite Element Methods for Pseudo-Hyperbolicequations

2011 ◽  
Vol 268-270 ◽  
pp. 908-912 ◽  
Author(s):  
Mei Xia Li
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Order Error ◽  
Expanded Mixed Finite Element ◽  
Physical Quantities ◽  
Element Method

H1-Galerkin mixed finite element method combining with expanded mixed element method are discussed for a class of second-order pseudo-hyperbolic equations. The methods possesses the advantage of mixed finite element while avoiding directly inverting the permeability tensor, which is important especially in a low permeability zone. Depended on the physical quantities of interest, the methods are discussed. The existence and uniqueness of numerical solutions of the scheme are derived and an optimal order error estimate for the methods is obtained.

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